Calculus IDerivatives

The chain rule, one layer at a time.

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Find the inside, differentiate the outside, then multiply by the inside’s derivative. Build the idea step by step, then try every question from your notes.

Inside and outside · Step 1 of 10

A composition feeds one output into another function

x\ \longrightarrow\ 2x} + {\displaystyle 1\ \longrightarrow\ (2x+1)^2

The order of operations

f(g(x))} = {\displaystyle (2x+1)^2

Write the composition

Inside and outside · Step 1 of 10

A composition feeds one output into another function

A function takes an input and returns an output. In

{f(g(x))}

, g is inside: do it first. Then feed that entire result into f, the outside. This is composition, not multiplication. Here

{f(x)} = {x^2}

and

{g(x)} = {2x+1}

Try it: Choose the order and move the input. Predict the two outputs before revealing them.

Order

Input x1

g(1)} = {\displaystyle 2(1)} + {\displaystyle 1} = {\displaystyle 3

First output

f(3)} = {\displaystyle (3)^2} = {\displaystyle 9

Final output

Check yourself

{\sqrt{1+\sin x}}

, what is the outermost operation?

Getting lost in the layers?

Bring your practice questions. We work through them together until every type feels routine.

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The chain rule, one layer at a time.

A composition feeds one output into another function

A composition feeds one output into another function

Which function goes first?

Your own function

Getting lost in the layers?

Questions, answered.

The chain rule, one layer at a time.

A composition feeds one output into another function

A composition feeds one output into another function

Which function goes first?

Your own function

Getting lost in the layers?

Questions, answered.

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